An alternative to direct solution of Laplacian equations, which
addresses the problem of boundary conditions both computationally and
conceptually, is the WKB method, which computes the leading
exponential dependence of probability distributions in large-system
limits. The key quantities in the WKB approximation are eikonals:
locally least-improbable trajectories for improbable excursions, along
which balance of probability currents determines the large-deviation
behavior of the
distribution.See [1]} for the
explanation that large-deviations scaling is the separation of scale
from structure in probability distributions for aggregate statistics.
Trajectories of eikonals are properties of the large-deviation rate
functions, while the probabilities associated with them are
exponential in the scale factors. Eikonal solutions for stationary
distributions, boundary values, and first-passage statistics of
diffusion solutions in basins of attraction were extensively developed
by Freidlin and Wentzell [2]}. Related
formulations in terms of 2-field functional integrals, for stochastic
processes on discrete state spaces including those of CRNs, were
derived by Doi and
Peliti [3]}, [4]}, [5]}, [6]}.
